Summary
In this paper, an Eisenmhart Model II with interaction for a GD-PBIB design withp replicates per cell is considered. Specifically the Model Yijl=µ+βi+τj+(βτ)ij+eijl is assumed, wherei=1, 2, ...,b; j=1, 2, ...,t andl=0, 1, 2, ...p s ij wheres ij=1, if treatmentj appears in blocki, 0, otherwise.
If βi, τj, (βτ)ij ande ijl are normally and independently distributed, then a minimal sufficient (Vector-valued) statistic for the class of densities for this model is found, together with the distribution of each component in the minimal sufficient statistic. It is also shown that the minimal sufficient statistic for this class densities is not complete. Hence the solution of the problem of finding minimum variance unbiased estimators of the variance components is not straightforward.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Afonja, B.: Minimal sufficient statistics for variance components for a general class of designs. Biometrika59 (2), 1972, 295–302.
Blackwell, D.: Conditional expectation and unbiased sequential estimation. Ann. Math. Stat.18 (March), 1947, 105–116.
Bose, R.C., andD.M. Mesner: On linear associative algebras corresponding to association scheme of pratially balanced designs. Ann. of Math. Stat.30 (December), 1959, 21–38.
Box, G.E.P.: Some theorems on quadratic forms — I. Ann. of Math. Stat.25 (June), 1954, 290–302.
Drygas, H.: Best quadratic unbiased estimation in variance-covariance component models. Math. Operationsfursch Statist. Ser. Statistics8 (2), 1977, 211–231.
Ferguson, Th.: Mathematical Statistics — A decision theoretical approach. New York 1967.
Hultquist, R.A., andF.A. Graybill: Minimal sufficient statistics for the two way classification mixed model design. Journal of the American Statistical Association60 (March), 1965, 182–192.
Kapadia, C.H.: Minimal sufficient statistics for the partically balanced incomplete block design with two associate classes under an Eisenhart Model II. Ann. of the Inst. of Stat. Math.14 (1), 1962, 63–71.
Kapadia, C.H., andD.L. Weeks: Variance components in two way classification model with interaction. Biometrika50 (December) 1963, 327–334.
Kapadia, C.H.: On the analysis of the group divisible designs. Journal of the American Statistical Association59 (December), 1964.
Kleffe, J.: Invariant methods for estimating variance components in mixed linear models. Math. Operationsfursch Statist. Ser. Statistics8 (2), 1977, 233–250.
Lehmann, E.L., andH. Scheffé: Completeness, similar regions and unbiased estimation. Sanhkyà10 Part 4, 1950, 305–340.
Low, L.Y.: Estimators of variance components in the balanced incomplete block design. Journal of the American Statistical Association64 (September), 1969, 1014–1030.
Neyman, J.: Sur un toerema concente le considdette statistiche sufficienti. Giornateddell Instituto Italrano Degli Attuari6, 1935, 320–333.
Neyman, J., andE.S. Pearson: Sufficient statistics and uniformly most powerful test of statistical hypotheses. Statistical Research Memoirs1–2, 1936, 113–117.
Pincus, R.: On tests in variance components models. Math. Operationsfursch. Statist. Ser. Statistics8 (2), 1977, 251–255.
Raghavarao, D.: Constructions and combinatorial problems in design of experiments. New York 1971.
Rao, C.R.: Minimum variance and the estimation of several parameters. Proceedings of the Cambridge Philosophical Society43, 1947, p. 280.
—: Estimation of heteroscedastic variances in linear models. J. Amer. Stat. Assoc.65 (March), 1970, 161–172.
Seely, J.: Minimal sufficient statistics and completeness for multi-variate normal families. Sankhyà SeriesA 39 Part 2, 1977, 170–185.
Weeks, D.L., andF.A. Graybill: Minimal sufficient statistics for the balanced incomplete block design under an Eisenhart Model II. Sankhyà SeriesA 23 Part 3, 1961, 261–268.
—: A minimal sufficient statistic for a general class of designs. Sankhyà SeriesA 24 Part 4, 1962, 339–354.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kapadia, C.H., Weeks, D.L. Minimal sufficient statistics for the group divisible partially balanced incomplete block design (GD-PBIBD) with interaction under an Eisenhart Model II. Metrika 31, 127–144 (1984). https://doi.org/10.1007/BF01915195
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF01915195